So shape \(A_n\) as described on the previous page is just one of many root polytopes! For our research purposes, we will only be considering the shapes \(A_n, B_n, C_n,\) \(and\) \(D_n\), since these shapes can take on any number of n dimensions.
\(\color{blue}{\text{Proposition:}}\) The root polytope \(B_n, C_n\) and \(D_n\) are balanced. The following polytopes are what \(B_3\), \(C_3\) and \(D_3\) look like.
We are only just starting to develop an understanding of the patterns of these root polytopes. The following patterns listed for the eigenvalues of polytopes \(B_n\), \(C_n\) and \(D_n\) are based off of our computations, and plans to complete proofs for each of these shapes is in the works!
The \(\color{green}{\text{spectrum}}\) of \(TL(B_n)\) is: \[ \frac{3\pm \sqrt{16n^2-40n+33}}{2}, \underbrace{0}_n, \underbrace{ \frac{2n+3\pm\sqrt{4n^2-20n+33}}{2}}_{n-1},\underbrace{2n-3}_{\frac{n(n-1)}{2}}, \underbrace{2n-1}_{n(n-1)}, \underbrace{2n+1}_{\frac{n(n-3)}{2}}\]
The \(\color{green}{\text{spectrum}}\) of \(TL(C_n)\) is \[\frac{(5-n)\pm \sqrt{9n^2-26n+33}}{2}, \underbrace{0}_n,\underbrace{ \frac{(5+n)\pm\sqrt{n^2-6n+33}}{2}}_{n-1}, \]\[\underbrace{n+1}_{n-1}, \underbrace{2(n-1)}_{\frac{n(n-1)}{2}}, \underbrace{2n}_{n(n-2)} ,\underbrace{2(n+1)}_{\frac{n(n-3)}{2}}\]
The \(\color{green}{\text{spectrum}}\) of \(TL(D_n)\) is \[2(2-n), \underbrace{4}_{n-1},\underbrace{0}_{n},\underbrace{2n}_{\frac{n(n-3)}{2}}, \underbrace{2(n-2)}_{\frac{n(n-1)}{2}},\underbrace{2(n-1)}_{n(n-2)}\]